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Right within Equilateral

In \(\triangle ABC\), \(AB= \dfrac{\sqrt{3}}{2}\), \(BC=1\) and \(\angle B = 90^{\circ}\).
\(PQR\) is an equilateral triangle with sides \(PQ\), \(QR\) and \(RP\) passing through the points \(A\), \(B\) an \(C\) respectively and each having length \(2\).

If the sum of the possible lengths of the segment \(BR\) be expressed as \( \dfrac{a}{b}\), where \(a\) , \(b\) are coprime positive integers, find the value of \(a+b\)